Solve -0.0164x^2 + 0.3655x + 25.151 = 0 for x.

Step-by-step solution

1. 1

   Notice how small the leading coefficient is. With $a=-0.0164$, $b=0.3655$, $c=25.151$, the quadratic term only becomes comparable to the constant when $x$ is in the tens:

   $|a|x^2\approx 25\ \Longrightarrow\ x\approx\sqrt{\tfrac{25}{0.0164}}\approx 39$

   So the roots should be large in magnitude — a useful expectation to hold before computing anything.

2. 2

   Compute the discriminant.

   $\Delta=0.3655^2-4(-0.0164)(25.151)=0.133590+1.649906=1.783496$

   $\sqrt{\Delta}=1.335476$

   Unusually, $b^2$ and $-4ac$ are of similar size here, because both $a$ and $b$ are small.

3. 3

   Apply the quadratic formula.

   $x=\frac{-0.3655\pm 1.335476}{2(-0.0164)}=\frac{-0.3655\pm 1.335476}{-0.0328}$

   Dividing by the very small $-0.0328$ is what magnifies a modest numerator into a large root.

4. 4

   Evaluate the two roots.

   $x_1=\frac{-0.3655+1.335476}{-0.0328}=-29.5724,\qquad x_2=\frac{-0.3655-1.335476}{-0.0328}=51.8590$

   The vertex sits at $x=-\tfrac{b}{2a}=\tfrac{0.3655}{0.0328}=11.143$, exactly midway between the roots, with maximum value $y\approx 27.19$.

5. 5

   Check with Vieta’s formulas. The sum must be $-\tfrac{b}{a}=\tfrac{0.3655}{0.0164}=22.2866$, and $-29.5724+51.8590=22.2866\;\checkmark$. The product must be $\tfrac{c}{a}=\tfrac{25.151}{-0.0164}=-1533.60$, and $(-29.5724)(51.8590)=-1533.60\;\checkmark$. Both checks are especially valuable here, since dividing by a small number amplifies any rounding error in the numerator.